Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,30,15,19,13,18)(11,20,22,16,24,32,23,33)(12,25,28,35,29,17,27,39)(21,36,37,40,38,34,26,31);(2,5)(3,7)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,32)(19,33)(20,30)(21,27)(25,36)(28,38)(29,37)(31,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,30)(15,19)(20,23)(21,31)(22,32)(24,33)(25,29)(26,34)(27,35)(28,39)(36,37)(38,40);(2,5)(6,8)(9,12,13,27,15,29,14,28)(10,17,18,35,19,25,30,39)(11,21,23,26,24,38,22,37)(16,31,20,34,33,40,32,36);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,30)(11,22,24,23)(12,27,29,28)(16,32,33,20)(17,35,25,39)(21,37,38,26)(31,36,40,34)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053032);